login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A322367
Number of disconnected or empty integer partitions of n.
3
1, 0, 1, 2, 3, 6, 7, 14, 17, 27, 34, 54, 63, 98, 118, 165, 207, 287, 345, 474, 574, 757, 931, 1212, 1463, 1890, 2292, 2898, 3515, 4413, 5303
OFFSET
0,4
COMMENTS
An integer partition is connected if the prime factorizations of its parts form a connected hypergraph. It is disconnected if it can be separated into two or more integer partitions with relatively prime products. For example, the integer partition (654321) has three connected components: (6432)(5)(1).
EXAMPLE
The a(3) = 2 through a(9) = 27 disconnected integer partitions:
(21) (31) (32) (51) (43) (53) (54)
(111) (211) (41) (321) (52) (71) (72)
(1111) (221) (411) (61) (332) (81)
(311) (2211) (322) (431) (432)
(2111) (3111) (331) (521) (441)
(11111) (21111) (421) (611) (522)
(111111) (511) (3221) (531)
(2221) (3311) (621)
(3211) (4211) (711)
(4111) (5111) (3222)
(22111) (22211) (3321)
(31111) (32111) (4221)
(211111) (41111) (4311)
(1111111) (221111) (5211)
(311111) (6111)
(2111111) (22221)
(11111111) (32211)
(33111)
(42111)
(51111)
(222111)
(321111)
(411111)
(2211111)
(3111111)
(21111111)
(111111111)
MATHEMATICA
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Sort[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n], Length[zsm[#]]!=1&]], {n, 20}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 04 2018
STATUS
approved